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Locally Quasi-Convex Compatible Topologies on a Topological Group

Faculty of Computer Science and Mathematics, Universität Passau, Innstr. 33, Passau D-94032, Germany
Department of Mathematics and Computer Science, University of Udine, Via delle Scienze, 208-Loc. Rizzi, Udine 33100, Italy
Instituto de Matemática Interdisciplinar y Departamento de Geometría y Topología, Universidad Complutense de Madrid, Madrid 28040, Spain
Author to whom correspondence should be addressed.
Academic Editor: Sidney A. Morris
Axioms 2015, 4(4), 436-458;
Received: 4 May 2015 / Revised: 28 September 2015 / Accepted: 8 October 2015 / Published: 13 October 2015
(This article belongs to the Special Issue Topological Groups: Yesterday, Today, Tomorrow)
PDF [336 KB, uploaded 13 October 2015]


For a locally quasi-convex topological abelian group (G,τ), we study the poset \(\mathscr{C}(G,τ)\) of all locally quasi-convex topologies on (G) that are compatible with (τ) (i.e., have the same dual as (G,τ) ordered by inclusion. Obviously, this poset has always a bottom element, namely the weak topology σ(G,\(\widehat{G})\) . Whether it has also a top element is an open question. We study both quantitative aspects of this poset (its size) and its qualitative aspects, e.g., its chains and anti-chains. Since we are mostly interested in estimates ``from below'', our strategy consists of finding appropriate subgroups (H) of (G) that are easier to handle and show that \(\mathscr{C} (H)\) and \(\mathscr{C} (G/H)\) are large and embed, as a poset, in \(\mathscr{C}(G,τ)\). Important special results are: (i) if \(K\) is a compact subgroup of a locally quasi-convex group \(G\), then \(\mathscr{C}(G)\) and \(\mathscr{C}(G/K)\) are quasi-isomorphic (3.15); (ii) if (D) is a discrete abelian group of infinite rank, then \(\mathscr{C}(D)\) is quasi-isomorphic to the poset \(\mathfrak{F}_D\) of filters on D (4.5). Combining both results, we prove that for an LCA (locally compact abelian) group \(G \) with an open subgroup of infinite co-rank (this class includes, among others, all non-σ-compact LCA groups), the poset \( \mathscr{C} (G) \) is as big as the underlying topological structure of (G,τ) (and set theory) allows. For a metrizable connected compact group \(X\), the group of null sequences \(G=c_0(X)\) with the topology of uniform convergence is studied. We prove that \(\mathscr{C}(G)\) is quasi-isomorphic to \(\mathscr{P}(\mathbb{R})\) (6.9). View Full-Text
Keywords: locally quasi-convex topology; compatible topology; quasi-convex sequence; quasi-isomorphic posets; free filters; Mackey groups locally quasi-convex topology; compatible topology; quasi-convex sequence; quasi-isomorphic posets; free filters; Mackey groups
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Außenhofer, L.; Dikranjan, D.; Martín-Peinador, E. Locally Quasi-Convex Compatible Topologies on a Topological Group. Axioms 2015, 4, 436-458.

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